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Valuation of Large Variable Annuity Portfolios withRank Order Kriging

Guojun Gan & Emiliano A. Valdez

To cite this article: Guojun Gan & Emiliano A. Valdez (2020) Valuation of Large Variable AnnuityPortfolios with Rank Order Kriging, North American Actuarial Journal, 24:1, 100-117, DOI:10.1080/10920277.2019.1617169

To link to this article: https://doi.org/10.1080/10920277.2019.1617169

Published online: 06 Sep 2019.

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Valuation of Large Variable Annuity Portfolios with Rank Order Kriging

Guojun Gan and Emiliano A. ValdezDepartment of Mathematics, University of Connecticut, Storrs, Connecticut

Metamodels, which simplify the simulation models used in the valuation of large variable annuity portfolios, have recentlyincreased in popularity. The ordinary kriging and the GB2 (generalized beta of the second kind) regression models are examplesof metamodels used to predict fair market values of variable annuity guarantees. It is well known that the distribution of fairmarket values is highly skewed. Ordinary kriging does not fit skewed data well but depends on only a few parameters that can beestimated straightforwardly. GB2 regression can handle skewed data but parameter estimation can be quite challenging. In thisarticle, we explore the rank order kriging method, which can handle highly skewed data and depends only on a single parameter,for the valuation of large variable annuity portfolios. Our numerical results demonstrate that the rank order kriging method per-forms remarkably well in terms of fitting the skewed distribution and producing accurate estimates of fair market values at theportfolio level.

1. INTRODUCTIONA variable annuity (VA) refers to an attractive insurance product that provides upside participation and downside protection

in both bull and bear markets. A main feature of variable annuities is that they contain guarantees (Hardy 2003; Ledlie et al.2008; The Geneva Association 2013). Due to the attractive guarantee features, lots of variable annuity contracts have beensold in the past decade. According to the Insured Retirement Institute, for example, new sales in the United States in 2016were 102 billion dollars and the total assets under management was 1.9 trillion.

Due to limited reinsurance capacity, dynamic hedging is widely adopted by insurance companies to mitigate the financialrisks arising from their VA business. In fact, the hedging programs helped to save the industry about $40 billion in Septemberand October 2008 during the global financial crisis (Chopra et al. 2009). However, dynamic hedging requires computingGreeks or sensitivities of the guarantees to major market factors. Because VA guarantees are relatively complex, their fair mar-ket values, which are used to calculate Greeks, cannot be evaluated explicitly. Insurance companies rely heavily on MonteCarlo simulation to calculate the fair market values. One major drawback of Monte Carlo simulation is that it is computation-ally intensive to value a large VA portfolio because every VA contract needs to be projected over many scenarios for a longtime horizon (Dardis 2016).

During the past few years, metamodeling approaches have been used to address the computational problem arising fromVA hedging. A metamodel is a surrogate model intended to simplify the original model. Using a metamodeling approach toestimate the fair market value (FMV) of a portfolio of VA contracts involves four major steps (Barton 2015):

1. Select a small number of representative VA contracts from the portfolio (experimental design).2. Run Monte Carlo simulation to calculate the FMVs of the selected VA contracts.3. Build a metamodel based on the selected VA contracts and the corresponding FMVs (predictive modeling).4. Use the metamodel to estimate the FMVs of all VA contracts in the portfolio.

As seen from the above steps, metamodeling techniques do not require valuing the full portfolio. Because the metamodel ismuch more computationally efficient than the Monte Carlo simulation model, it substantially reduces the valuation time.

Address correspondence to Guojun Gan, Department of Mathematics, University of Connecticut, 341 Mansfield Road, Storrs, CT06268-1009. E-mail: guojun.gan@uconn.edu

100

North American Actuarial Journal, 24(1), 100–117, 2020# 2019 Society of ActuariesISSN: 1092-0277 print / 2325-0453 onlineDOI: 10.1080/10920277.2019.1617169

The metamodeling approach depends on two major components: an experimental design and a predictive model. Researchto date has focused on these two components. Gan (2013) used the k-prototypes clustering algorithm for the experimentaldesign and ordinary kriging as the metamodel. Gan and Lin (2015) used the k-prototypes algorithm to select representative VAcontracts and universal kriging for functional data as the metamodel. Because the k-prototypes algorithm is not efficient forselecting a moderate number (e.g., 200) of representative VA contracts, Latin hypercube sampling (LHS; Gan 2015) and thetruncated fuzzy c-means algorithm (Gan and Huang 2017) have been proposed to select representative contracts. In particular,Gan and Valdez (2016) investigated several experimental design methods and found that clustering and LHS are comparablein accuracy and are better than other methods such as random sampling. Gan and Lin (2017) used the LHS and conditionalLHS methods with ordinary and universal kriging to calculate dollar deltas quickly for daily hedging purpose.

Hejazi and Jackson (2016) and Xu et al. (2018) used neural networks as metamodels for the valuation of large VAportfolio. Hejazi, Jackson and Gan (2017) treated the valuation of large VA portfolios as a spatial interpolation problemand studied several interpolation methods, including inverse distance weighting and radial basis function. Gan andValdez (2017a) studied the use of a copula to model the dependency of partial dollar deltas and found that using acopula does not improve the prediction accuracy of the metamodel because the dependency is well captured by thecovariates. Gan and Valdez (2018) proposed the use of the GB2 (generalized beta of the second kind) distribution tomodel the fair market values in order to address the skewness typically observed in the distribution of the fair marketvalues. Gan (2018) studied the use of linear models with interaction terms as metamodels for the valuation of largeVA portfolios.

Among the aforementioned metamodels, ordinary kriging and the GB2 regression model are two popular choices. Theadvantage of ordinary kringing over the GB2 regression model is that the former depends only on a few parameters that can beestimated straightforwardly. However, one drawback of ordinary kriging is that the dependent variable (i.e., the fair marketvalue) is assumed to follow a Gaussian distribution. This assumption is not appropriate for the fair market value of the guaran-tees as the distribution of fair market values is positively skewed. Although the GB2 regression model addresses the skewnessof the dependent variable, estimating the parameters of the GB2 regression model posed additional challenges (Gan andValdez 2018).

In this article, we study the use of rank order kriging (Pachepsky, Radcliffe, and Selim 2003) to model the fair market val-ues of VA guarantees. Rank order kriging is also known as quantile kriging. This method estimates the rank order of thedependent variable given the values of the independent variables and transforms the estimated rank order to the original scale.Rank order kriging has the advantages that it depends only on a few parameters and that it is effective for handling highlyskewed data (Juang, Lee, and Ellsworth 2000).

The remainder of this article is organized as follows. In Section 2, we give a description of the data we use to demonstratethe usefulness of rank order kriging for VA valuation. In Section 3, we introduce and describe the rank order kriging methodin detail. In Section 4, we present some numerical results to show the performance of rank order kriging. Section 5 concludesthe article with some remarks.

2. DESCRIPTION OF THE DATAWe use the synthetic data set produced in Gan and Valdez (2017b) to evaluate the performance of the rank order kriging

method. This data set consists of a portfolio of 190,000 VA policies and their fair market values. Each VA policy is character-ized by 45 features. Some features (e.g., fund fees) have identical values for all policies. We exclude these features and use theremaining features as explanatory variables. The following list contains the variables we use for modeling the fair mar-ket values:

� gender: Gender of the policyholder� productType: Product type of the VA contract� gmwbBalance: Guaranteed Minimum Withdrawal Benefit (GMWB) balance� gbAmt: Guaranteed benefit amount� FundValuei: Account value of the ith fund, for i ¼ 1, 2, . . . , 10� age: Age of the policyholder. It is calculated from the birth date and the valuation date.� ttm: Time to maturity in years. It is calculated from the valuation date and the maturity date.

Among the variables listed above, gender and productType are the only categorical variables. The portfolio contains 76,007(about 40%) policies with female policyholders. There are 19 product types and each product type has 10,000 policies.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 101

Table 1 shows some summary statistics of the continuous variables. From the table, we see that except for age and ttm, allvariables are dollar amounts and have zeros. Table 2 shows some summary statistics of the fair market values. Many policieshave negative fair market values because for these policies the benefit is less than the risk charge. From Table 2 we also seethat the mean is much higher than the median, indicating that the distribution of the fair market values is positively skewed.The histogram shown in Figure 1 confirms the skewness of the distribution.

The fair market values shown above were calculated by Monte Carlo simulation, which is computationally intensive. Asreported in Gan and Valdez (2017b), it would take a single CPU (central processing unit) about 108 hours to calculate the fairmarket values for all of the policies in the portfolio.

3. RANK ORDER KRIGINGIn this section, we describe the rank order kriging method in detail. Figure 2 shows a high-level procedure of this method.

From the figure, we see that in the rank order kriging method, standardized ranks are estimated and back transformed to getthe fair market values in the original scale.

3.1. Ordinary Kriging for RanksTo describe the ordinary kriging method for ranks, let z1, z2, :::, zk be the k representative VA policies and let v1, v2, :::, vk

be the corresponding fair market values (or other quantities of interest). For j ¼ 1, 2, :::, k, let uj be the standardized rank of vj;that is,

uj ¼ uðzjÞ ¼ rðvjÞk

, (1)

TABLE 1Summary Statistics of the Continuous Variables

Min. 1st Quarter Median Mean 3rd Quarter Max.

gmwbBalance 0.00 0.00 0.00 35,611.54 0.00 499,708.73gbAmt 0.00 186,864.95 316,225.98 326,834.59 445,940.63 1,105,731.57FundValue1 0.00 0.00 12,635.17 33,433.87 49,764.15 1,099,204.71FundValue2 0.00 0.00 15,107.17 38,542.81 56,882.55 1,136,895.87FundValue3 0.00 0.00 10,043.96 26,740.18 39,199.69 752,945.34FundValue4 0.00 0.00 10,383.79 26,141.80 39,519.79 610,579.68FundValue5 0.00 0.00 9,221.26 23,026.50 35,023.00 498,479.36FundValue6 0.00 0.00 13,881.41 35,575.67 52,981.06 1,091,155.87FundValue7 0.00 0.00 11,541.47 29,973.25 44,465.70 834,253.63FundValue8 0.00 0.00 11,931.41 30,212.11 45,681.16 725,744.64FundValue9 0.00 0.00 11,562.79 29,958.29 44,302.35 927,513.49FundValue10 0.00 0.00 11,850.05 29,862.24 44,967.78 785,978.60age 34.52 42.03 49.45 49.49 56.96 64.46ttm 0.59 10.34 14.51 14.54 18.76 28.52

TABLE 2Summary Statistics of the Fair Market Values

Min. 1st Quarter Median Mean 3rd Quarter Max.

fmv �94.94 �5.14 12.49 67.59 66.81 1,536.70

Note: Numbers are in thousands.

102 G. GAN AND E. A. VALDEZ

where rðvjÞ 2 f1, 2, :::, kg is the rank order of vj; that is, rðvjÞ is the position of vj when v1, v2, :::, vk are arranged in ascendingorder. The standardized rank transformation is monotonically increasing.

Let x1, x2, :::, xn be the VA policies in a portfolio, where n is the number of VA policies in the portfolio. Under an ordinarykriging model, the standardized rank of the fair market value of the guarantees embedded in the ith policy xi is assumed to be(Cressie 1993; Juang, Lee, and Ellsworth 2000):

UðxiÞ ¼ lþ dðxiÞ, i ¼ 1, 2, :::, n,

where l is an unknown constant and dð�Þ is a zero-mean intrinsically stationary spatial process. In this model, UðxiÞ can bepredicted as (Cressie 1993)

uðxiÞ ¼Xmj¼1

wijuðzjÞ, (2)

where uðzjÞ is the standardized rank order of the jth representative policy as defined in Equation (1) and wi1,wi2, :::,wim are thekriging weights. These kriging weights are obtained by solving the following linear equation system:

V11 � � � V1m 1... . .

. ... ..

.

Vm1 � � � Vmm 11 � � � 1 0

0BBB@

1CCCA �

wi1

..

.

wim

hi

0BBB@

1CCCA ¼

Di1

..

.

Dim

1

0BBB@

1CCCA, (3)

FIGURE 1. A Histogram of the Fair Market Values. Note: Numbers in the horizontal axis are in thousands.

FIGURE 2. Sketch of the Rank Order Kriging Method.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 103

where hi is the Lagrange multiplier to ensure the sum of the kriging weights equal to one and Vls and Dil are semivariogramsthat describe the degree of spatial dependence of the fair market values. Mathematically, Vls and Dil are calculated as

Vls ¼ cðjjzl�zsjjÞ, (4a)

Dil ¼ cðjjxi�zljjÞ, (4b)

where jj � jj is the L2 norm (i.e., the Euclidean distance) and cðhÞ is a semivariogram function defined as (Chiles and Delfiner2012):

cðhÞ ¼ 12Var uðz1Þ�uðz2Þ½ �, h ¼ jjz1�z2jj:

There are several theoretical semivariogram models, including linear, spherical, exponential, and Gaussian models. For example,the linear semivariogram model can be specified as (Chiles and Delfiner 2012)

cðhÞ ¼ bh,

where b is a parameter. A linear model is appropriate when spatial variability increases linearly with distance. The exponentialsemivariogram model can be specified as (Isaaks and Srivastava 1990)

cðhÞ ¼ 1� exp � 3bh

� �, l, s ¼ 1, 2, :::,m,

where b>0 is a parameter.To select a suitable theoretical semivariogram model, we investigate the empirical semivariogram given by (Juang, Lee,

and Ellsworth 2000)

ceðhÞ ¼1

2jSðhÞjX

ðs, tÞ2SðhÞ½uðsÞ�uðtÞ�2, (5)

where S(h) is a set of all pairs of policies that have a distance of h; that is,

SðhÞ ¼ fðs, tÞ : jjs�tjj ¼ h, s 2 X0, t 2 X0g,

and jSðhÞj is the number of elements in S(h). Here X0 ¼ fz1, z2, :::, zkg is the set of representative policies.

3.2. Back TransformationMost of the rank order kriging estimates uðxiÞ are between 0 and 1. There are a few estimates that are outside of this inter-

val. We need to transform these kriging estimates back to the original scale. This can be done by an interpolation model.Before transforming the rank order kriging estimates back to the scale of the original data, we need to correct the smoothingeffect of these estimates (Yamamoto 2010).

A uniform random variable on an interval (a, b) has an expected value of aþb2 and a variance of ðb�aÞ2

12 . Because the standar-dized rank order UðxÞ has a uniform distribution on ð1=n, 1Þ, it has a variance of 1

12 ðn�1n Þ2, which is approximately 1

12 when n islarge. However, the empirical variance of the rank order kriging estimates uðx1Þ, :::, uðxnÞ is usually less than 1

12 due to thesmoothing effect of the kriging method. Yamamoto (2005) proposed a procedure to correct the smoothing effect caused bykriging. This procedure involves cross-validation, which is time consuming. In this article, we use a different and simpleapproach to correct the smoothing effect.

104 G. GAN AND E. A. VALDEZ

In particular, we modify the rank order kriging estimates as follows:

u�ðxiÞ ¼ rðuðxiÞÞn

, (6)

where rðuðxiÞÞ is the position of uðxiÞ when uðx1Þ, uðx2Þ, :::, uðxnÞ are arranged in ascending order. Since

fu�ðxiÞ : i ¼ 1, 2, :::, ng ¼�1n,2n, :::, 1

�,

the empirical variance of u�ðx1Þ, :::, u�ðxnÞ is equal to 112 ðn�1

n Þ2. In addition, the modified rank order kriging estimates fallwithin the interval ð0, 1�, whereas the raw estimates may fall outside the interval.

Then we can transform the modified rank order kriging estimates back to the original scale. This can be done by using aregression model or an interpolation method. For simplicity, we use a linear interpolation method given as follows:

y�i ¼ vi1 þu�ðxiÞ�ui1ui2 � ui1

ðvi2�vi1Þ, (7)

where i1 and i2 are indices such that ui1 and ui2 are the ranks closest to u�ðxiÞ. The estimated fair market values y�1, y�2, :::, y

�n

usually contain biases due to the data transformation (Garcia et al. 2010). As a result, we need to adjust the biases in these esti-mates to get the final estimates of the fair market values. We use the multiplicative bias adjustment method (Garcia et al.2010) to adjust the biases as follows:

yi ¼ cy�i , (8)

where c ¼ llm

is the ratio of an estimate of the mean of the fair market values over the mean of the fair market values obtainedfrom the kriging model; for example,

l ¼ 1k

Xki¼1

vi, lm ¼ 1n

Xni¼1

y�i :

This back-transformation approach, although ad hoc, is efficient and works well for our purpose as demonstrated by thenumerical results in this article.

4. NUMERICAL RESULTSIn this section, we present some numerical results to demonstrate the performance of the rank order kriging method. In par-

ticular, we compare the rank order kriging method with the ordinary kriging method and the GB2 regression model.

4.1. Experimental SetupMetamodeling has two major components: an experimental design method and a metamodel. The experimental design

method is used to select representative VA contracts. The metamodel is used to predict the fair market values of all of the VAcontracts in the portfolio. In this article, our focus is to develop metamodels. In order to compare the performance of metamo-dels in a consistent manner, we use the same experimental design method for all metamodels. Gan and Valdez (2016) com-pared several experiment design methods and found that clustering-based methods produce good results. In our numericalexperiments, we use the hierarchical k-means algorithm (Gan and Valdez 2019) to select representative VA policies. InAppendix B we provide numerical results of the metamodels for two additional experimental design methods.

Another important factor to consider in experimental design is the number of representative VA policies. Intuitively, thenumber of representative VA policies cannot be too large or too small. Using a too large number of representative VA policieswill increase the runtime significantly because all representative VA policies have to be valued by Monte Carlo simulation.Using a too small number of representative VA policies is also not appropriate because the resulting metamodel may not pro-duce accurate predictions. In previous studies (e.g., Gan and Lin 2015; Gan 2018), the number of representative VA contracts

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 105

is determined to be 10 times the number of predictors, including the dummy variables converted from categorical variables. Inthis article, we follow this strategy to determine the number of representative VA policies. Because there are 34 predictors, westart with 340 representative VA contracts. We also use 170 representative VA contracts to see the impact of the number ofrepresentative VA contracts on the performance of the metamodels.

4.2. Validation MeasuresTo compare the performance of the metamodels in terms of accuracy, we use the following four validation meas-

ures: the percentage error, the mean error, the R2 (Frees 2009), and the concordance correlation coefficient of quantiles(Lin 1989). The first two measures assesses the accuracy of the metamodels at the portfolio level. The third measureassesses the accuracy at the individual policy level. The last measure is used to assess the agreement between twoempirical distributions.

Let yi and yi denote the fair market value of the ith VA policy in the portfolio that is calculated by Monte Carlo simulationand estimated by a metamodel, respectively. Then the percentage error (PE) and the mean error (ME) are defined as follows:

PE ¼Pn

i¼1ðyi � yiÞPni¼1yi

, (9)

ME ¼Pn

i¼1ðyi � yiÞn

, (10)

where n is the number of VA policies in the portfolio. The lower the absolute values of the PE and the ME, the betterthe results.

The R2 is defined as follows:

R2 ¼ 1�Pn

i¼1ðyi�yiÞ2Pni¼1ðyi��yÞ2

, (11)

where �y is the average of y1, y2, :::, yn. The lower the R2, the better the results.

The concordance correlation coefficient for quantiles (CCCQ) is used to measure the agreement between the quantiles oftwo samples. To define this measure, we let qj and qj be the 100j

m th quantile of fy1, y2, :::, yng and fy1, y2, :::, yng, respectively,for j ¼ 1, 2, :::,m, where m is a positive integer (e.g., m¼ 1000). Then CCCQ is defined as follows (Lin 1989):

CCCQ ¼ 2qr1r2r21 þ r22 þ ðl1�l2Þ2

, (12)

where q is the correlation between ðq1, q2, :::, qmÞ and ðq1, q2, :::, qmÞ, r1 and l1 are the standard deviation and the mean ofðq1, q2, :::, qmÞ, respectively, and r2 and l2 are the standard deviation and the mean of ðq1, q2, :::, qmÞ, respectively. A higherCCCQ means more agreement between the quantiles of the fair market values obtained from two models.

4.3. ResultsWe applied the rank order kriging method, the ordinary kriging method, and the GB2 regression model to predict the fair

market values with k¼ 340 and k¼ 170 representative VA policies. Table 3 shows the accuracy of the three models when 340representative VA policies were used. Except for the R2, all validation measures show that rank order kriging performs the

TABLE 3Accuracy of the Metamodels When k¼ 340 Representative VA Policies Were Used

PE ME R2 CCCQ

Rank order kriging 0.0018 0.1236 0.8121 0.9958Ordinary kriging 0.0032 0.2191 0.8009 0.9557GB2 regression 0.0382 2.5811 0.8227 0.9853

106 G. GAN AND E. A. VALDEZ

best among the three models. For example, the percentage error obtained by rank order kriging is 0.18%, which is much lowerthan that obtained by the GB2 regression model. In terms of R2, the GB2 regression model performs the best because it pro-duced the highest R2.

From Table 3, we also see that rank order kriging produced a CCCQ value of 0.9958, which is close to 1. This indicatesthat the distribution of the fair market values estimated by rank order kriging matches quite well the distribution of the fairmarket values calculated by Monte Carlo.

Figure 3 shows the scatterplot and the Q–Q plot of the fair market values estimated by rank order kriging and those calcu-lated by Monte Carlo when 340 representative VA policies were used. The scatter plot shows that the estimation at the individ-ual policy level is not very accurate. However, the points scatter quite symmetrically around the 45

�line. The Q–Q plot shows

that the rank order kriging method did a good job of fitting the skewed data. The points fall closely to the 45�line.

Figures 4 and 5 show the scatter and Q–Q plots obtained from the ordinary kriging method and the GB2 regression model,respectively. The scatterplot in Figure 4(a) shows that the estimates obtained from ordinary kriging are biased. The Q–Q plotin Figure 4(b) shows that ordinary kriging does not fit the tail well. Figure 5(a) shows that the scatterplot produced by GB2

FIGURE 3. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Rank Order Kriging When k¼ 340Representative VA Policies Were Used.

FIGURE 4. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Ordinary Kriging When k¼ 340Representative VA Policies Were Used.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 107

regression is quite symmetric around the 45�line. The Q–Q plot in Figure 5(b) shows that GB2 regression fits the tail well

although the fit is a little bit off in the middle.Comparing Figures 3, 4, and 5, we see that the rank order kriging method works pretty well in terms of fitting skewed data.

The ordinary kriging method does not fit well the tail of skewed data as expected. Table 4 shows some summary statistics ofthe fair market values estimated by the three methods as well as those calculated by Monte Carlo. From the table, we see thatsummary statistics obtained from the rank order kriging match well those obtained from Monte Carlo.

Figures 6(a) and 6(b) shows the empirical semivariograms of rank order kriging and ordinary kriging that are estimatedfrom the data, respectively. From the figures, we see that the empirical semivariogram of ordinary kriging has a wide range.Both empirical semivariograms show that a linear theoretical semivariogram can be used. However, the ordinary krigingmethod does not work with the linear semivariogram due to the wide range of semivariogram values. Instead, we used theexponential semivariogram for the ordinary kriging method.

Figure 7(a) and (b) shows the histograms of the rank order kriging estimates and the modified rank order kriging estimates,respectively. Figure 7(a) shows that the standardized ranks estimated by rank order kriging have bell shapes and a reduced vari-ance due to the smoothing effect of the kriging method. In fact, the variance of the rank order kriging estimates is around 0.066,which is less than the variance of a uniform variable on ð1=n, 1Þ. Figure 7(b) shows that the modified rank order kriging estimateshave a uniform distribution.

Table 5 shows the accuracy of the three models when 170 representative VA policies were used. Again, except for the R2,all validation measures indicate that the rank order kriging method performs the best among the three models. The R2 valuesshow that the rank order kriging method does not fit well the data at the individual policy level. However, the CCCQ valueshows that rank order kriging fits the distribution pretty well.

FIGURE 5. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by GB2 Regression When k¼ 340Representative VA Policies Were Used.

TABLE 4Summary Statistics of the Fair Market Values Estimated by Different Methods

When k¼ 340 Representative VA Policies Were Used

Min. 1st Quarter Median Mean 3rd Quarter Max.

Monte Carlo –94.94 –5.14 12.49 67.59 66.81 1,536.70Rank order kriging –43.54 –5.02 11.88 67.71 59.90 1,244.11Ordinary kriging –50.38 6.97 28.62 67.81 70.67 1,210.32GB2 regression –29.24 –4.10 19.68 70.17 57.23 1,459.24

Note: Numbers are in thousands.

108 G. GAN AND E. A. VALDEZ

Figures 8, 9, and 10 show the scatter and QQ plots produced by the three models when 170 representative VA policies wereused. The QQ plots in Figures 8(b), 9(b), and 10(b) show that the rank order kriging method fit the skewed data quite well.However, the scatterplot in Figure 8(a) shows that the fit produced by rank order kriging is not very accurate at the individualpolicy level. The rank order kriging method overestimated and underestimated the fair market values of some policies. Thenumber of overestimates is comparable to the number of underestimates. As a result, the scatterplot is quite symmetric aroundthe 45� line. Table 6 shows some summary statistics of the fair market values obtained from the three methods when 170 rep-resentative VA policies were used. The numbers in the table show that rank order kriging performed the best in terms ofmatching the quantiles.

Figure 11 shows the empirical semivariograms of the rank order kriging method and the ordinary kriging method when 170representative VA policies were used. We see patterns similar to those observed before and the semivariograms can be

FIGURE 6 (a) Empirical Semivariograms of Rank order Kriging and (b) Ordinary Kriging When k¼ 340 Representative VA Policies Were Used.

FIGURE 7. (a) Histograms of the Rank Order Kriging Estimates and (b) Modified Rank Order Kriging Estimates When k¼ 340 Representative VA PoliciesWere Used.

TABLE 5Accuracy of the Metamodels When k¼ 170 Representative VA Policies Were Used

PE ME R2 CCCQ

Rank order kriging 0.0040 0.2729 0.2381 0.9947Ordinary kriging 0.0865 5.8491 0.6061 0.8568GB2 regression –0.0073 –0.4951 0.8097 0.9784

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 109

modeled linearly. However, the linear semivariogram for ordinary kriging has a wide range of values and does not work forordinary kriging. Again, we used the exponential semivariogram for ordinary kriging. Figure 12 shows the histograms of therank order kriging estimates and the modified estimates. We also see patterns similar to those observed before. The distributionof the rank order kriging estimates has a bell shape and a reduced variance.

Finally, Table 7 shows the runtime of the three models. The runtime includes the time used to fit the model and the timeused to predict the fair market values. The time used to select representative policies is not included. From the table, we seethat the GB2 regression model is the fastest model among the three models. Rank order kriging and ordinary kriging are muchslower than GB2 regression due to the large number of distance calculations. However, the runtime used by rank order krigingwith k¼ 340 was around 5minutes, which is much less than the 108 hours used by Monte Carlo simulation (Gan andValdez 2017b).

In summary, the numerical results show that the rank order kriging method is able to fit quite well the distribution of fairmarket values that is highly skewed. In addition, the rank order kriging method produces accurate estimates at the portfoliolevel and its runtime is similar to that of ordinary kriging.

FIGURE 8. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Rank Order Kriging When k¼ 170Representative VA Policies Were Used.

FIGURE 9. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Ordinary Kriging When k¼ 170Representative VA Policies Were Used.

110 G. GAN AND E. A. VALDEZ

FIGURE 10. Scatter and Q–Q plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by GB2 Regression When k¼ 170Representative VA Policies Were Used.

TABLE 6Summary Statistics of the Fair Market Values Estimated by Different Methods

When k¼ 170 Representative VA Policies Were Used

Min. 1st Quarter Median Mean 3rd Quarter Max.

Monte Carlo –94.94 –5.14 12.49 67.59 66.81 1536.70Rank order kriging –93.01 –5.68 11.83 64.63 65.66 1228.46Ordinary kriging –68.37 2.12 25.77 68.40 76.07 964.12GB2 regression –37.47 –3.56 18.07 66.12 67.27 1098.31

Note: Numbers are in thousands.

FIGURE 11. (a) Empirical Semivariograms of Rank Order Kriging and (b) Ordinary Kriging When k¼ 170 Representative VA Policies Were Used.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 111

5. CONCLUSIONSThe ordinary kriging and the GB2 regression model are two popular metamodels used to predict the fair market values of

VA guarantees. The advantage of ordinary kringing over the GB2 regression model is that the former is less demanding interms of parameter estimation. However, ordinary kriging assumes that the dependent variable (i.e., the fair market value) fol-lows a Gaussian distribution. This assumption is not appropriate for the fair market value of the guarantees because the distri-bution of fair market values is positively skewed. Although the GB2 regression model addresses the skewness of thedependent variable, estimating the parameters of the GB2 regression model is quite challenging (Gan and Valdez 2018).

In this article, we studied the use of rank order kriging to predict the fair market values of VA guarantees. Rank order krig-ing utilizes data transformation and thus is able to handle highly skewed data. Our numerical results show that the rank orderkriging method works well as expected. In particular, we compared rank order kriging with GB2 regression and ordinary krig-ing and found that rank order kriging is better than GB2 regression and ordinary kriging in terms of fitting highly skewed data.

FUNDINGThe authors acknowledge the financial support provided by the Committee on Knowledge Extension Research (CKER) of

the Society of Actuaries.

ORCIDGuojun Gan http://orcid.org/0000-0003-3285-7116

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FIGURE 12. (a) Histograms of the Rank Order Kriging Estimates and (b) Modified Rank Order Kriging Estimates When k¼ 170 Representative VA PoliciesWere Used.

TABLE 7Runtime of the Metamodels

Rank order kriging Ordinary kriging GB2 regression

k¼ 340 291.75 286.62 6.38k¼ 170 129.12 129.29 3.33

Note: Numbers are in seconds.

112 G. GAN AND E. A. VALDEZ

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Discussions on this article can be submitted until October 1, 2020. The authors reserve the right to reply to any discussion.Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

APPENDIX A. GB2 REGRESSION MODELThe GB2 regression model was proposed to predict the fair market values of VA guarantees (Gan and Valdez 2018). A

GB2 random variable has the following probability density function (Cummins et al. 1990):

f ðxÞ ¼ jajbBðp, qÞ

xb

� �ap�1

1þ xb

� �a" #�p�q

, x>0, (A.1)

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 113

where a 6¼ 0, p> 0, q> 0, b> 0, and B(p, q) is the beta function. The parameters a, p, and q are referred to as the shapeparameters of the GB2 distribution. The parameter b is called the scale parameter. When �p< 1

a<q, the expectation existsand is given by

E X½ � ¼ bB pþ 1a , q� 1

a

� �Bðp, qÞ : (A.2)

Let Y denote the fair market value of guarantees embedded in a VA policy. Because Y can be negative, the shifted fairmarket value

X ¼ Y þ c

is modeled with a GB2 distribution, where c is the shift parameter to be estimated from the data.Independent or regressor variables are incorporated through the scale parameter b in order to make sure that the expecta-

tions can be calculated for all policies. The method of maximum likelihood is used to estimate the parameters. Let s be thenumber of VA policies in the experimental design. For i ¼ 1, 2, :::, s, let vi be the fair market value of the guaranteesembedded in the ith VA policy in the experimental design. Then the log-likelihood function of the model is defined asfollows:

LðhÞ ¼ s lnjaj

Bðp, qÞ�apXs

i¼1

zTi bþ ðap�1ÞXs

i¼1

ln ðvi þ cÞ

�ðpþ qÞXs

i¼1

ln 1þ vi þ cexp ðzTi bÞ

� �a" #

,(A.3)

where h ¼ ða, p, q, c, bÞ and zi is a numerical vector representing the ith VA policy in the experiment design. It is worth not-ing that estimating the parameters of the GB2 regression model is quite challenging. Gan and Valdez (2018) proposed amulti-stage procedure to estimate the parameters.

The fitted GB2 regression model is used to predict the fair market values of guarantees for the portfolio. Let n be thenumber of VA policies in the portfolio and xi the numeric vector representing the ith VA policy in the portfolio. Then thefair market value of guarantees for the ith VA policy can be estimated as follows:

yi ¼exp ðxTi bÞB p þ 1

a , q � 1a

� �Bðp, qÞ �c, i ¼ 1, 2, :::, n, (A.4)

where a, p, q, c, and b are parameters estimated from the data.

APPENDIX B. COMPARISON OF EXPERIMENTAL DESIGNSIn this appendix, we provide some test results about the sensitivity of the experimental design methods on the three meta-

models. Gan and Valdez (2016) compared five experimental design methods for the GB2 regression model and found thatthe data clustering method and the conditional Latin hypercube sampling method produce the most accurate results. Herewe compare two additional experimental design methods—conditional Latin hypercube sampling and random sampling—forthe three metamodels.

Table B.1 shows the performance of the three metamodels based on the two experimental design methods. The resultsare mixed. Table B.1(a) shows that GB2 regression works the best when conditional Latin hypercube sampling was used toselect representative policies. Table B.1(b) shows that rank order kriging produced the best result in terms of CCCQ whenthe random sample was used. In terms of PE and ME, rank order kriging did not perform as well as the other two metamo-dels when conditional Latin hypercube sampling and random sampling were used to select representative policies. In termsof R2 and CCCQ, rank order kriging outperformed ordinary kriging.

Figure B.1 shows the scatter plot and the Q–Q plot produced by rank order kriging with conditional Latin hypercubesampling. These plots indicate that rank order kriging does not work well with conditional Latin hypercube sampling.Figure B.2 shows the scatterplot and the QQ plot produced by rank order kriging with random sampling. Comparing

114 G. GAN AND E. A. VALDEZ

TABLE B.1Accuracy of the Metamodels When k¼ 340 Representative VA Policies Were Selected by the

Two Different Experimental Design Methods

(a) Conditional Latin hypercube sampling

PE ME R2 CCCQ

Rank order kriging –0.0720 –4.8676 0.8156 0.9791Ordinary kriging –0.0365 –2.4654 0.7965 0.9256GB2 regression 0.0331 2.2375 0.8238 0.9971(b) Random sampling PE ME R2 CCCQRank order kriging –0.0335 –2.2673 0.8396 0.9954Ordinary kriging 0.0117 0.7876 0.8099 0.9520GB2 regression 0.0172 1.1600 0.8729 0.9887

FIGURE B.2. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Rank Order Kriging When k¼ 340Representative VA Policies Were Selected by Random Sampling.

FIGURE B.1. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Rank Order Kriging When k¼ 340Representative VA Policies Were Selected by Conditional Latin Hypercube Sampling.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 115

FIGURE B.5. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by GB2 Regression When k¼ 340Representative VA Policies Were Selected by Conditional Latin Hypercube Sampling.

FIGURE B.3. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Ordinary Kriging When k¼ 340Representative VA Policies Were Selected by Conditional Latin Hypercube Sampling.

FIGURE B.4. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by Ordinary Kriging When k¼ 340Representative VA Policies Were Selected by Random Sampling.

116 G. GAN AND E. A. VALDEZ

Figure B.2 with Figure B.1, we see that rank order kriging produced better results when random sampling was used toselect representative policies.

Figures B.3, B.4, B.5, and B.6 show the scatterplots and the Q–Q plots produced by ordinary kriging and GB2 regressionwith the two experimental design methods. These figures show that ordinary kriging does not fit the right tail well and GB2regression produced better results in both cases.

FIGURE B.6. Scatter and Q–Q Plots of the Fair Market Values Calculated by Monte Carlo and Those Estimated by GB2 Regression When k¼ 340Representative VA Policies Were Selected by Random Sampling.

VALUATION OF LARGE VARIABLE ANNUITY PORTFOLIOS 117